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Mirrors > Home > NFE Home > Th. List > mp3an3 | GIF version |
Description: An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
Ref | Expression |
---|---|
mp3an3.1 | ⊢ χ |
mp3an3.2 | ⊢ ((φ ∧ ψ ∧ χ) → θ) |
Ref | Expression |
---|---|
mp3an3 | ⊢ ((φ ∧ ψ) → θ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mp3an3.1 | . 2 ⊢ χ | |
2 | mp3an3.2 | . . 3 ⊢ ((φ ∧ ψ ∧ χ) → θ) | |
3 | 2 | 3expia 1153 | . 2 ⊢ ((φ ∧ ψ) → (χ → θ)) |
4 | 1, 3 | mpi 16 | 1 ⊢ ((φ ∧ ψ) → θ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∧ w3a 934 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 |
This theorem depends on definitions: df-bi 177 df-an 360 df-3an 936 |
This theorem is referenced by: mp3an13 1268 mp3an23 1269 mp3anl3 1273 vfinnc 4471 ov 5595 ovmpt2a 5714 ovmpt2 5716 enmap1lem5 6073 ltcpw1pwg 6202 nnltp1c 6262 nnc3n3p1 6278 |
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